From 06b50c9b063a425da16fda3e88e3742371ff7d70 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Mon, 11 May 2026 21:22:27 -0400 Subject: [PATCH] Adjusted statement of FTC for path integrals. --- src/fa/rs/path.tex | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/src/fa/rs/path.tex b/src/fa/rs/path.tex index 3d8699a..4f045a9 100644 --- a/src/fa/rs/path.tex +++ b/src/fa/rs/path.tex @@ -77,12 +77,14 @@ \begin{theorem}[Fundamental Theorem of Calculus for Path Integrals] \label{theorem:ftc-path-integrals} - Let $[a, b], [c, d] \subset \real$, $E, F$ be separated locally convex spaces, $\gamma \in C([a, b]; F)$ be a rectifiable path, $U \in \cn_F(\gamma([a, b]))$. + Let $[a, b] \subset \real$, $E, F$ be separated locally convex spaces, $\gamma \in C([a, b]; F)$ be a rectifiable path, $U \in \cn_F(\gamma([a, b]))$. Let $\sigma \subset \mathfrak{B}(F)$ be an ideal containing all compact sets, then for any $f \in C^1_\sigma(U; E)$, \[ \int_\gamma D_\sigma f = f(\gamma(b)) - f(\gamma(a)) \] + + In particular, if $\gamma(a) = \gamma(b)$, then $\int_\gamma D_\sigma f = 0$. \end{theorem} \begin{proof} Using \autoref{lemma:rectifiable-piecewise-linear}, assume without loss of generality that $\gamma$ is piecewise smooth. By the \hyperref[Chain Rule]{proposition:chain-rule-sets}, $f \circ \gamma \in C^1([a, b]; F)$ with $D(f \circ \gamma)(t) = Df(\gamma(t)) \cdot D\gamma(t)$. In which case, by \autoref{proposition:lebesgue-stieltjes-differentiable} and the \hyperref[Fundamental Theorem of Calculus]{theorem:ftc-riemann},